NOTES ON D-MODULES AND CONNECTIONS WITH HODGE THEORY DONU ARAPURA These notes are almost entirely expository and result from my attempt to learn this material. The first part summarizes D-module theory up to Riemann-Hilbert. The second part discusses vanishing cycles in its various forms. This is needed in the next part which summarizes the basics of Morihiko Saito’s theory of Hodge modules. My main motivation for going through all this was to convince myself that Saito’s methods and the more naive construction in [A] yield the same mixed Hodge structure on the cohomology of a geometric variation of Hodge structure. The proof of this is given in part 4. Readers interested in just this part, may find the note [A2] on the“ArXiv” more convenient. I gave some informal talks on this material at KIAS in Seoul in 2005 and TIFR Mumbai in 2008. I would them for giving me the opportunity to do so. Contents 1. D-modules 2 1.1. Weyl Algebra 2 1.6. Dn-modules 3 1.22. Inverse and direct image 5 1.29. Differential operators on affine varieties 7 1.35. Non-affine varieties 8 1.39. Connections 10 1.43. Riemann-Hilbert correspondence 11 1.47. Perverse Sheaves 12 1.56. Gauss-Manin connections 13 2. Vanishing cycles 15 2.1. Vanishing cycles 15 2.3. Perverse Sheaves on a disk 16 2.6. Kashiwara-Malgrange filtration 17 3. Hodge modules 19 3.1. Hodge theory background 19 3.6. Filtered D-modules 20 3.7. Hodge modules on a curve 21 3.11. VHS on a punctured disk 22 3.18. Hodge modules: introduction 23 3.24. Hodge modules: conclusion 24 3.26. Mixed Hodge modules 25 3.30. Explicit construction 27 3.32. Real Hodge modules 28 4. Comparison of Hodge structures 29 1 2 DONU ARAPURA References 30 1. D-modules 1.1. Weyl Algebra. Fix a positive integer n. The nth Weyl algebra Dn over C is the ring of differential operators with complex polynomial coefficients in n variables. More formally, Dn can be defined as the noncommutative C-algebra generated by ∂ ∂ symbols x1, . xn, ∂1 = , . ∂n = subject to relations ∂x1 ∂xn xixj = xjxi ∂i∂j = ∂j∂i ∂ixj = xj∂i, if i 6= j ∂ixi = xi∂i + 1 The last two relations stem from the Leibnitz rule ∂i(xjf) = ∂i(xj)f +xj∂if. These relations can be expressed more succinctly, using commutators as [xi, xj] = [∂i, ∂j] = 0 [∂i, xj] = δij There is a sense in which Dn is almost commutative that I want to explain. From the defining relations, it follows that any P ∈ Dn can be expanded uniquely as X I J P = αI,J x ∂ n I I1 In where I,J ∈ N , x = x1 . xn etc. The maximum value of J1 + ...Jn occurring in this sum is the order of P . We write FkDn for the space of operators of order at most k. It is easy to see that FkFm ⊆ Fk+m. Thus the associated graded M Gr(Dn) = Fk/Fk−1 k inherits a graded algebra structure. Lemma 1.2. Given P, Q ∈ Dn, we have order([P, Q]) < order(P ) + order(Q) Sketch. It’s enough to check this when P, Q are monomials, i.e. expressions of the form xI ∂J . In this case, it is a straight forward consequence of induction and the defining relations. Corollary 1.3. Gr(Dn) is commutative. Slightly more work yields: Theorem 1.4. Gr(Dn) is isomorphic to the polynomial ring C[x1, . xn, ξ1, . ξn] = R2n I want to sketch a slightly nonstandard proof of this. First “quantize” Dn to obtain a ring Hn which has an additional variable q subject to the relations that q commutes with xi and ∂j and [∂i, xj] = qδij The remaining relations are the same as for Dn: the x’s and ∂’s commute among themselves. I will call Hn the Heisenberg algebra, since it nothing but the universal enveloping algebra of the Heisenberg Lie algebra. We see from the relations that NOTES ON D-MODULES AND CONNECTIONS WITH HODGE THEORY 3 Dn = Hn/(q−1) and the “classical limit” Hn/(q) is the polynomial ring R2n, where (q − λ) is the two sided ideal generated by this element. Now form the Rees algebra M k Rees = t Fk ⊂ C[t] ⊗ Dn with t a central element. The theorem will follow from the next result which is straightforward. Lemma 1.5. ∼ (1) Gr(Dn) = Rees/(t) (2) The map Rees → Hn sending xi 7→ xi, t∂j 7→ ∂j and t 7→ q is an isomor- phism. In more geometric terms, we have an identification between Gr(Dn) and the ring of polynomial functions on the cotangent bundle T ∗Cn. 1.6. Dn-modules. The notion of a Dn-module gives an abstract way to think about systems of linear partial differential equations in n-variables. Since the ring Dn is noncommutative, we have to be careful about distinguishing between left- modules and right Dn-modules. I will often be lazy, and refer to a left Dn-module simply as a Dn-module. Here are some examples. Example 1.7. Dn is automatically both a left and right Dn-module. Example 1.8. Let Rn = C[x1, . xn] be the polynomial ring. This is a left Dn- ∂ module where xi acts by multiplication, and ∂i by . ∂xi P Example 1.9. Given operators P1 ...PN ∈ Dn, the left (resp. right) ideal DnPi P (resp. PiDn) are left (resp. right) Dn-modules. Likewise for the quotients P P P Dn/ DnPi (resp. Dn/ PiDn). Note that Rn = Dn/ Dn∂i. −1 −1 Example 1.10. Given a nonzero polynomial, Rn[f ] = C[x1, . xn, f ] is a Dn-module, where the derivatives act by differentiation of rational functions. Example 1.11. Let F be any space of complex valued functions on Cn which is an algebra over the polynomial ring and and closed under differentiation, then it ∞ becomes a left Dn-module as above. In particular, this applies to holomorphic, C and C∞ functions with compact support. Example 1.12. The space of distributions is the topological dual of the space of C∞ functions with compact support or test functions. This has a right module structure defined a follows. Given a distribution δ, a test function φ and P ∈ Dn, let δP (φ) = δ(P φ). The first four examples above are finitely generated. (The last example requires −N some thought. In the special case f = x1, we see immediately that x1 can be obtained by repeated differentiation.) Fix a space of functions F as in example 1.11. Given a left Dn-module M, define the space of solutions by Sol(M) = HomDn (M, F ) To justify this terminology consider the example 1.9 above. We see immediately that there is an exact sequence P P 0 → Sol(M) → F −→i F N 4 DONU ARAPURA Therefore Sol(M) is the space of solutions of the system Pi(f) = 0. There is a symmetry between right and left modules that I will refer to as “flipping”. ∗ ∗ Lemma 1.13. There is an involution P 7→ P of Dn determined by xi = xi and ∗ ∗ ∂i = −∂i. Given a right Dn-module M, the operation P · m = mP makes M into a left module, which I denote by F lipR→L(M). This gives an equivalence between the categories of (finitely generated) left and right modules. The inverse operation will denote by F lipL→R. Suppose that M is a finitely generated Dn-module. We define good filtration on M to be a filtration FpM such that (1) The filtration FpM = 0 for p 0 and ∪FpM = M. (2) Each FpM is a finitely generated Rn-submodule. (3) FpDn · FqM ⊆ Fp+qM. Lemma 1.14. Every finitely generated module possess a good filtration. N Proof. Write it as a quotient of some free module Dn and take the image of N (FpDn) The filtration is not unique, however it does lead to some well defined invariants. Given a module with good filtration, the associate graded M Gr(M) = FpM/Fp−1M is a finitely generated Gr(Dn)-module. The annihilator of Gr(M) gives an ideal ∼ in Gr(Dn) = R2n. The zero set of this ideal defines a subvariety Ch(M, F ) ⊂ C2n called the characteristic variety or singular support. Since Gr(M) is graded with respect to the natural grading on Gr(Dn), we see that, the annihilator is homogeneous. Therefore ∗ Lemma 1.15. Ch(M, F ) is invariant under the action of t ∈ C by (xi, ξj) 7→ (xi, tξj). We can view this another way. Consider the Rees module Rees(M, F ) = ⊕FpM. This is a finitely generated module over Hn such that Gr(M) = Hn/(q)⊗Rees(M, F ). So in some sense Ch(M, F ) is the classical limit of M as q → 0. Theorem 1.16. Ch(M, F ) is independent of the filtration. Thus we can, and will, drop F from the notation. Example 1.17. In the previous examples, we see that 2n (1) The annihilator of Gr(Dn) is 0, so that Ch(Dn) = C . P (2) Taking Rn = Dn/ Dn∂i, yields Gr(Rn) = C[x1, . xn]. Its annihilator n is the ideal (ξ1, . ξn). Therefore Ch(Rn) = C × 0 −1 −1 (3) Let M = R1[x ], where we x = x1. Then 1, x generate M. Let −1 −k−1 FkM = Fk · 1 + Fk · x = C[x]x This gives a good filtration.